Back to the Table of Contents

Statistical Probabilities and Distributions - Lesson 12

Another Distribution: The Student t

Lesson Overview

Student t Distribution

It is often the case that one wants to calculate the size of sample needed to obtain a certain level of confidence in survey results. Unfortunately, this calculation requires prior knowledge of the population standard deviation ([sigma]). Realistically, [sigma] is unknown. Often a preliminary sample will be conducted so that a reasonable estimate of this critical population parameter can be made. If such a preliminary sample is not made, but confidence intervals for the population mean are to be constructing using an unknown [sigma], then the distribution known as the Student t distribution can be used.

First, a little history about this curious name. William Gosset (1876-1937) was a Guinness Brewery employee who needed a distribution that could be used with small samples. Since the Irish brewery did not allow publication of research results, he published under the pseudonym of Student. We know that large samples approach a normal distribution. What Gosset showed was that small samples taken from an essentially normal population have a distribution characterized by the sample size. The population does not have to be exactly normal, only unimodal and basically symmetric. This is often characterized as heap-shaped or mound shaped.

Following are the important properties of the Student t distribution.

  1. The Student t distribution is different for different sample sizes.
  2. The Student t distribution is generally bell-shaped, but with smaller sample sizes shows increased variability (flatter). In other words, the distribution is less peaked than a normal distribution and with thicker tails. As the sample size increases, the distribution approaches a normal distribution. For n > 30, the differences are negligible.
  3. The mean is zero (much like the standard normal distribution).
  4. The distribution is symmetrical about the mean.
  5. The variance is greater than one, but approaches one from above as the sample size increases ([sigma]2=1 for the standard normal distribution).
  6. The population standard deviation is unknown.
  7. The population is essentially normal (unimodal and basically symmetric)

To use the Student t distribution which is often referred to just as the t distribution, the first step is to calculate a t-score. This is much like finding the z-score. The formula is:

t = ([x bar] - µ) ÷ (s/sqrt(n))

Actually, since the population mean is likely also unknown, often the t-score will be looked up based on the level of confidence desired and the degrees of freedom and the population estimated. Degrees of freedom is a fairly technical term which permeates all of inferential statistics. In this case, it is n-1.

In general, the degrees of freedom is the number of values that can vary
after certain restrictions have been imposed on all values.

Where does the term degrees of freedom come from? Suppose, for example, that you have a phone bill from Ameritech that says your household owes $100. Your mother and father state that $70 of it is theirs and that your younger sibling owes only $5. How much does that leave you? Here, n=3 (parents, sibling, you), but once you have the total (or mean) and two more pieces of information, the last data element is constrained. The same is true with the degrees of freedom, you can arbitrarily use any n-1 data points, but the last one will be determined for a given mean. Another example is with 10 tests that averaged 55, if you assign nine people random grades, the last test score is not random, but constrained by the overall mean. Thus for 10 tests and a mean, there are nine degrees of freedom.

If the interval calls for a 90% confidence level, then alpha = 0.10 and alpha/2 = 0.05 (for a two-tailed test). Tables of t values typically have a column for degrees of freedom and then columns of t values corresponding with various tail areas. An abbreviated table is given below. For a complete set of values consult a larger table or your TI-83+ graphing calculator. DISTR 5 gives tcdf. tcdf expects three arguments, lower t value, upper t value, and degrees of freedom. Since no inverse t function is given on the calculator, some guessing may be involved. Note how tcdf(9.9,9E99,2) indicates a t value of about 9.9 for a one tailed area of 0.005 with two degrees of freedom. Please locate the corresponding value of 9.925 in the table.

As with other confidence intervals, we use the t-score to obtain the margin of error term which is added and subtracted from the statistic of interest (in this case, the sample mean) to obtain a confidence interval for the parameter of interest (in this case, the population mean). In this case the margin of error is defined (since you don't have population standard deviation you use the sample's) as:

ME = talpha/2 • (s ÷ sqrt(n))

Your confidence interval should look like: [x bar] - ME < µ < [x bar] + ME.

Table of t Values

The headings in the table below, such as .005/.01 indicate the left/right tail area (0.005) for a one tail test or the total tail area (left+right=0.01) for a two tailed test. In general, if an entry for the degrees of freedom you desire is not present in the table, use an entry for the next smaller value of the degrees of freedom. This guarantees a conservative estimate.

Degrees of Freedom\1/2 tails .005/.01.01/.02.025/.05.05/.10.10/.20
163.6631.8212.716.3143.078
29.9256.9654.3032.9201.886
35.8414.5413.1822.3531.638
44.6043.7472.7762.1321.533
54.0323.3652.5712.0151.476
103.1692.7642.2281.8121.372
152.9472.6022.1321.7531.341
202.8452.5282.0861.7251.325
252.7872.4852.0601.7081.316
z 2.5752.3261.9601.6451.282

Although the t procedure is fairly robust, that is it does not change very much when the assumptions of the procedure are violated, you should always plot the data to check for skewness and outliers before using it on small samples. Here small can be interpreted as n < 15. If your sample is small and the data is clearly nonnormal or outliers are present, do not use the t. If your sample is not small, but n < 40, and there are outliners or strong skewness, do not use the t. Since the assumption that the samples are random is more important that the normality of the population distribution, the t statistic can be safely used even when the sample indicates the population is clearly skewed, if n > 40.

Two sample t Tests

Often one wants to compare two treatments or populations and determine if there is a difference. This can be done either with or without matching. We will discuss first the unmatched situation. Two assumptions are used: two independent simple random samples from two distinct populations (matching would negate independence); and both populations are normally distributed with unknown means and standard deviations. Our null hypothesis would look like H0: µ1=µ2 or we might want to give a confidence interval for the difference µ1-µ2. We use the sample means and standard deviations to estimate the unknown parameters. Although the statistic [x bar]1 - [x bar]2 has a normal distribution in terms of the combined population variance, when we use the combined sample variance, we do not obtain a t distribution. Nonetheless, we do use the t distribution for hypothesis testing in this case. The two-sample t statistic is as follows:
t = (([x bar]1 - [x bar]2) - (µ1 - µ2)) ÷ sqrt(s12/n1 + s22/n2)

The expression in the denominator reflects the way variances sum (standard deviations do not sum). There are two options for obtaining a value for the degrees of freedom. Calculate a fractional degrees of freedom as given below, or use the smaller of n1-1 or n2-1. This latter value always results in conservative results. As sample size increases, this latter procedure also becomes more accurate. The two-sample t procedures are more robust than the one-sample methods, especially when the distributions are not symmetric. If the sizes of the two samples are equal and the two distributions have similar shapes, it can be accurate down to sample sizes as small as n1 = n2 = 5. The two-sample t procedure is most robust against nonnormality when the two samples are of equal size. Thus when planning such a study, you should make them equal.

The fractional degrees of freedom formula is as follows:
d.f.= (s12/n1 + s22/n2) ÷ (((s12/n1)2 ÷ (n1-1)) + ((s22/n2)2 ÷ (n2-1)))

Matched Pair Test

Comparitive studies are more convincing than single sample investigations. Thus one sample inference testing is less common. A common design compares two treatments, either before and after, or randomly picking one of each pair for treatment. In a such a matched pair design, we apply the one sample t procedures to the observed differences. Our null hypothesis would be that these differences are zero and our alternative hypothesis would be that they are not (two-tailed) or positive/negative (one-tailed).

An example might be before and after SAT scores after a high-priced course of study. Or your typical freshman practice EXPO project where peas, corn, or other seeds are grown with and without (control) a treatment. Some Biology instructions and EXPO judges have expected our freshmen to perform this calculation!

T. OF CONTENTS HOMEWORK SOLUTIONS ACTIVITY CONTINUE